Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 2 - Differentiation - 2.3 Exercises - Page 127: 88

Answer

x = $\frac{3\pi}{4}$ and x = $\frac{7\pi}{4}$

Work Step by Step

The rate of change of a function is the same as its derivative. Thus, this question is asking us when the derivative of f(x) and g(x) are equal. f(x) = sec(x) g(x) = csc(x) The derivative of sec(x) = secxtanx The derivative of csc(x) = -cscxcotx Setting these equal to eachother, we get sec(x)tan(x) = -csc(x)cot(x) $\frac{sec(x)tan(x)}{csc(x)cot(x)}$ = -1 Taking the derivative of this, we get $\frac{\frac{1}{cos(x)}\times\frac{sin(x)}{cos(x)}}{\frac{1}{sin(x)}\times\frac{cos(x)}{sin(x)}}$ = -1 Simplifying this, we get $\frac{(sin(x))^{3}}{(cos(x))^{3}}$ = -1. $\frac{sin(x)}{cos(x)}$ = tan(x), thus $\frac{(sin(x))^{3}}{(cos(x))^{3}}$ = (tan(x))$^{3}$ = -1. tan(x) = $\sqrt[3] -1$ = -1 The only x values for which this statement is true on the interval (0, 2$\pi$] are x = $\frac{3\pi}{4}$ and x = $\frac{7\pi}{4}$
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